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Institut f¨ur Informatik JLU Gießen Arndtstraße 2 D-35392 Giessen, Germany Tel: +49-641-99-32141 Fax: +49-641-99-32149 mail@informatik.uni-giessen.de www.informatik.uni-giessen.deInstitut f¨
ur Informatik
Tight Bounds on the
Descriptional Complexity
of Regular Expressions
Hermann Gruber Markus Holzer
IFIG Research Report 0901 February 2009
Justus-Liebig-Universit¨
at
Gießen
IFIG Research Report
IFIG Research Report 0901, February 2009
Tight Bounds on the Descriptional Complexity of
Regular Expressions
Hermann Gruber1 and Markus Holzer2 Institut f¨ur Informatik, Universit¨at Giessen
Arndtstraße 2, D-35392 Giessen, Germany
Abstract.We improve on some recent results on lower bounds for conversion problems for regular expressions. In particular we consider the conversion of planar deterministic finite automata to regular expressions, study the effect of the complementation operation on the descriptional complexity of regular expressions, and the conversion of regular expressions extended by adding intersection or interleaving to ordinary regular expressions. Almost all obtained lower bounds are optimal, and the presented examples are over a binary alphabet, which is best possible.
Categories and Subject Descriptors: F.1.1 [Computation by Abstract Devices]: Models of Computation—Relations between models; F.1.2 [Computation by Abstract Devices]: Modes of Computation—Parallelism and concurrency; F.3.1 [Logics and Meanings of Programs]: Specifying and Verifying and Reasoning about Programs—Specification tech-niques; F.4.3 [Mathematical Logic and Formal Languages]: Formal Languages— Operations on languages
Additional Key Words and Phrases: Interleaving, shuffle, semi-extended regular expressions, regular expressions
1E-mail: hermann.k.gruber@informatik.uni-giessen.de 2E-mail: markus.holzer@informatik.uni-giessen.de
Conversion known results this paper with |Σ| = 2 planar DFA to RE 2Θ(√n) for |Σ| = 4 [10] 2Θ(√n) [Thm. 4]
22Ω(√n log n) for |Σ| = 2 [10] ¬ RE to RE 22Ω(n) for |Σ| = 4 [8] 2 2Θ(n) [Thm. 8] RE( ∩ ) to RE 22Ω(√n) for |Σ| = 2 [7] 22Θ(n) [Thm. 9] 22Ω(n/ log n) [Thm. 16] RE( x ) to RE 22Ω(√n) for |Σ| const. [7]
22Θ(n) for |Σ| = O(n) [Thm. 10] Table 1.Comparing the lower bound results for conversion problems of deterministic finite automata (DFA), regular expressions (RE), and regular expressions with additional operations (RE(·)), where ∩ denotes intersection, ¬ complementation, and x the interleaving or shuffle operation on formal languages. Entries with a bound in Θ(·) indicate that the result is best possible, i.e., refers to a lower bound matching a known upper bound.
1 Introduction
It is well known that regular expressions are equally expressive as finite au-tomata. In contrast to this equivalence, a by now classical result due to Ehren-feucht and Zeiger states that finite automata, even deterministic ones, can some-times allow exponentially more succinct representations than regular expres-sions [4]. Although they obtained a tight lower bound on expression size, their examples used a largely growing alphabet.
Reducing the alphabet size remained an open challenge [5] until the recent advent of new proof techniques, see [8, 10, 13]—most of our proofs in this pa-per rely on the recently established relation between regular expression size and star height of regular languages [10]. Although this resulted in quite a few new insights into the nature of regular expressions,see also [7, 11, 12], proving tight lower bounds for small alphabets remains a challenging task, and not all bounds in the mentioned references are both tight and cover all alphabet sizes. In this work, we close some of the remaining gaps: In the case of converting planar finite automata to regular expressions, we reduce the alphabet size to binary while retaining the tight lower bound. We prove this directly, by finding a witness language over a binary alphabet. For the other questions under consideration, namely the effect of complementation and of extending regular expression syn-tax by adding an intersection or interleaving operator, proceeding in this way appears more difficult. Yet, sometimes it proves easier to find witness languages over larger alphabets. For this case, we also devise a new set of encodings which are economic and, in some precise sense, robust with respect to both the Kleene star and the interleaving operation. This extends the outreach of known proof techniques, and allows us to give a definitive answer to some questions regard-ing the descriptional complexity of regular expressions that were not yet settled completely in previous works [5, 7, 8, 10]. Our main results are summarized and compared to known results in Table 1.
2 Basic Definitions
We introduce some basic notions in formal language and automata theory—for a thorough treatment, the reader might want to consult a textbook such as [16]. In particular, let Σ be a finite alphabet and Σ∗ the set of all words over the alphabet Σ, including the empty word ǫ. The length of a word w is denoted by |w|, where |ǫ| = 0. A (formal) language over the alphabet Σ is a subset of Σ∗.
The regular expressions over an alphabet Σ are defined recursively in the usual way:1 ∅, ǫ, and every letter a with a ∈ Σ is a regular expression; and
when r1 and r2 are regular expressions, then (r1+ r2), (r1· r2), and (r1)∗ are
also regular expressions. The language defined by a regular expression r, denoted by L(r), is defined as follows: L(∅) = ∅, L(ǫ) = {ǫ}, L(a) = {a}, L(r1+ r2) =
L(r1) ∪ L(r2), L(r1 · r2) = L(r1) · L(r2), and L(r1∗) = L(r1)∗. The size or
alphabetic width of a regular expression r over the alphabet Σ, denoted by alph(r), is defined as the total number of occurrences of letters of Σ in r. For a regular language L, we define its alphabetic width, alph(L), as the minimum alphabetic width among all regular expressions describing L.
Our arguments on lower bounds for the alphabetic width of regular guages is based on a recent result that utilizes the star height of regular lan-guages [10]. Here the star height of a regular language is defined as follows: For a regular expression r over Σ, the star height, denoted by h(r), is a struc-tural complexity measure inductively defined by: h(∅) = h(ǫ) = h(a) = 0, h(r1· r2) = h(r1 + r2) = max (h(r1), h(r2)), and h(r1∗) = 1 + h(r1). The star
height of a regular language L, denoted by h(L), is then defined as the minimum star height among all regular expressions describing L. The next theorem estab-lishes the aforementioned relation between alphabetic width and star height of regular languages [10]:
Theorem 1. Let L ⊆ Σ∗ be a regular language. Then alph(L) ≥ 213(h(L)−1)− 1.
The star height of a regular language appears to be more difficult to deter-mine than its alphabetic width, see, e.g., [14]. Fortunately, the star height can be determined more easily for a certain subclass of regular languages, namely the family of bideterministic regular languages, which are defined as follows: A regular language L is bideterministic if there exists a deterministic finite au-tomaton A with a single final state such that a deterministic finite auau-tomaton accepting the reversed language LR is obtained from A by reverting the direc-tion of each transidirec-tion and exchanging the roles of the initial and final state. For these languages, the star height can be determined from the digraph structure of the minimal DFA: The cycle rank of a digraph G = (V, E), denoted by cr(G), is inductively defined as follows: (1) If G is acyclic, then cr(G) = 0. (2) If G is strongly connected, then cr(G) = 1 + minv∈V{cr(G − v)}, where G − v denotes the graph with the vertex set V \{v} and appropriately defined edge set. (3) If G
1 For convenience, parentheses in regular expressions are sometimes omitted and the
con-catenation is simply written as juxtaposition. The priority of operators is specified in the usual fashion: concatenation is performed before union, and star before both product and union.
is not strongly connected, then cr(G) equals the maximum cycle rank among all strongly connected components of G.
For a given finite automaton A, let its cycle rank, denoted by cr(A), be defined as the cycle rank of the underlying digraph. Eggan’s Theorem states that the star height of a regular language equals the minimum cycle rank among all NFAs accepting it [3]. Later, the following was proved by McNaughton in [19], building on his earlier work [20]:
Theorem 2 (McNaughton’s Theorem). Let L be a bideterministic language, and let A be the minimal trim, i.e., without a dead state, deterministic finite au-tomaton accepting L. Then h(L) = cr(A).
In fact, the minimality requirement in the above theorem is not needed, since every bideterministic finite automaton in which all states are useful is already a trim minimal deterministic finite automaton. Here, a state is useful if it is both reachable from the start state, and if some final state is reachable from it.
3 Lower Bounds on Regular Expression Size
This section is three folded. First we show an optimal bound converting pla-nar deterministic finite automata to equivalent regular expressions and then we present our results on the alphabetic width on complementing regular ex-pression and on regular exex-pressions with intersection and interleaving. While the former result utilizes a characterization of cycle rank in terms of a cops and robbers game given in [10], the latter two results are mainly based on star height preserving homomorphisms.
3.1 Converting Planar DFAs into Regular Expressions
For the main result of this subsection we need a characterization of cycle rank in terms of a cops and robber game [20, 10]. This characterization provides a useful tool in proving lower bounds on the cycle rank of specific families of digraphs. The cops and strong visible robber game, defined in [17], is given as follows: Let G= (V, E) be a digraph. Initially, the cops occupy some set of X ⊆ V vertices, with |X| ≤ k, and the robber is placed on some vertex v ∈ V \ X. At any time, some of the cops can reside outside the graph, say, in a helicopter. In each round, the cop player chooses the next location X′ ⊆ V for the cops. The stationary
cops in X ∩X′remain in their positions, while the others go to the helicopter and fly to their new position. During this, the robber player, knowing the cops’ next position X′ from wire-tapping the police radio, can run at great speed to any
new position v′, provided there is both a (possibly empty) directed path from v to v′, and a (possibly empty) directed path back from v′ to v in G − (X ∩ X′), i.e., he has to avoid to run into a stationary cop, and to run along a path inside the current strongly connected component of the graph induced by the vertices free of stationary cops. Afterwards, the helicopter lands the cops at their new positions, and the next round starts, with X′ and v′ taking over the roles of X
and v, respectively. The cop player wins the game if the robber cannot move any more, and the robber player wins if the robber can escape indefinitely.
The immutable cops variant of the above game restricts the movements of the cops in the following way: Once a cop has been placed on some vertex of the graph, he has to stay there forever. The hot-plate variant of the game restricts the movements of the robber in that he has to move along a nontrivial path in each move—even if the path consists only of a self-loop. The following theorem from [10] gives a characterization of the cycle rank in terms of such a game. Theorem 3. Let G be a digraph and k ≥ 0. Then k cops have a winning strategy for the immutable cops and hot-plate strong visible robber game if and only if the cycle rank of G is at most k.
Now we are ready for the main result of this subsection. Note that in [5] it was shown that for planar finite automata, one can construct equivalent regular expressions of size at most 2O(√n), for all alphabet sizes polynomial in n. This
is a notable improvement over the general case, since conversion from n-state deterministic finite automata to equivalent regular expressions was shown to be of order 2Θ(n)in [10]. Also in [10] a tight lower bound on the conversion of planar
deterministic finite automata to regular expressions of 2Θ(√n)for alphabet size at least four was proven. Next we improve this result to alphabets of size two, using the above given characterization of cycle rank in terms of a cops and robber game.
Theorem 4. There is an infinite family of languages Lnover a binary alphabet
acceptable by n-state planar deterministic finite automata, such that alph(Ln) =
2Ω(√n).
Fig. 1.A drawing of the graph G3. When viewed as automaton A3, the solid (dashed,
respec-tively) arrows indicate a-transitions (b-transitions, respecrespec-tively).
Proof. By Theorems 1 and 2, it suffices to find an infinite family of bidetermin-istic finite automata Ak of size O(k2) such that the digraph underlying Ak has
cycle rank Ω(k).
The deterministic finite automata Ak witnessing the claimed lower bound
are inspired by a family of digraphs Gkdefined in [17]. These graphs each admit
a planar drawing as the union of k concentric equally directed 2k-cycles, which are connected to each other by 2k radial directed k-paths, the first k of which are directed inwards, while the remaining k are directed outwards; see Figure 1
for illustration. Formally, for k ≥ 1, let Gk = (V, E) be the graph with vertex
set V = { ui,j | 1 ≤ i, j ≤ k } ∪ { vi,j | 1 ≤ i, j ≤ k }, and whose edge set can
be partitioned into a set of directed 2k-cycles Ci, and two sets of directed
k-paths Piand Qiwith 1 ≤ i ≤ k. Here each Ciadmits a walk visiting the vertices
ui,1, ui,2, . . . , ui,k, vi,1, vi,2, . . . , vi,k in order, each Pi admits a walk visiting the
vertices u1,i, u2,i, . . . , uk,i in order, and Qi admits a walk visiting the vertices
vk,i, vk−1,i, . . . , vk,1 in order.
Fix {a, b} as a binary input alphabet. If we interpret the edges in Gk
be-longing to the cycles Ci as a-transitions, the edges belonging to the paths Pi
and Qias b-transitions, interpret the vertices as states and choose a single initial
and a single final state (both arbitrarily), we obtain a finite automaton Ak with
O(k2) states whose underlying digraph is Gk. It is easily observed that Ak is
bideterministic; thus it only remains to show that for the underlying graph Gk
holds cr(Gk) = Ω(k).
To this end, we use the game characterization of cycle rank given by Theo-rem 3, by showing that the robber can escape against k cops in the immutable cops and hot-plate strong visible robber game. It was shown in [17] that on the graph Gk, the robber has a winning strategy against k cops in the cops and
strong visible robber game, even if the cops are not immutable and can freely jump between vertices on the graph. The result is, however, not established for the hot-plate variant of the game, which restricts the allowed movements of the robber. But note that at most one additional cop is needed if we drop the hot-plate restriction: The allowed movements of the robber in hot-plate variant coincide with the original ones as long as he resides in a strongly connected component of size at least two. The situation becomes different only once the robber is finally trapped in a strongly connected component consisting only of one vertex (and no loop). When playing the game in the hot-plate variant, the robber is caught in this situation. Otherwise, we place the additional cop at this vertex to catch the robber. Note that since the extra cop never moves, this argument equally applies in the immutable cops variant of the game. ⊓⊔ 3.2 Operations on Regular Expressions: Alphabetic Width of
Complementation
As noted in [5], the naive approach to complement regular expressions, of con-verting first the given expression into an nondeterministic finite automaton, determinizing, complementing the resulting deterministic finite automaton, and converting back to a regular expression gives a doubly exponential upper bound of 22O(n). The authors of [5] also gave a lower bound of 2Ω(n), and stated as an open problem to find tight bounds. A doubly-exponential lower bound was found in [8], but only for alphabets of size at least four. Their witness language is a 4-symbol encoding of the set of walks in an n-vertex complete digraph. They gave a very short regular expression describing the complement of the encoded set, and provided a direct and technical proof showing that the encoded language requires large regular expressions, carefully adapting the approach originally taken by Ehrenfeucht and Zeiger [4]. Resulting from an independent approach
pursued by the authors, in [10] a roughly doubly-exponential lower bound of 22O(√n log n) was given for binary alphabet.
Now it appears tempting to encode the language from [8] using a star height preserving homomorphism to further reduce the alphabet size, as done in [10] for a similar problem. Unfortunately, the proof from [8] does not offer any clue about the star height of the witness language, and thus we cannot mix these proof techniques. At least, it is known [2] that the preimage of the encoded language has large star height:
Theorem 5 (Cohen). Let Jn be the complete digraph on n vertices with
self-loops, where each edge (i, j) carries a unique label aij. Let Wn denote the set
of all walks ai0i1ai1i2· · · air−2ir−1air−1ir in Jn, including the empty walk ǫ. Then
the star height of language Wn equals n.
To obtain a tight lower bound for binary alphabets, here we use a similar encoding as in [8], but make sure that the encoding is a star height preserv-ing homomorphism. Here a homomorphism ρ preserves star height, if the star height of each regular language L equals the star height of the homomorphic image ρ(L). The existence of such encodings was conjectured in [3] and proved in [20]:
Theorem 6. Let Σ = {a1, a2, . . . , ad} be a finite alphabet with d ≥ 1 and define
σ: Σ∗ → {a, b}∗ be the homomorphism given by σ(a
i) = aibd−i+1, for 1 ≤ i ≤ d.
Then for every regular language L over Σ, the star height of L equals the star height of σ(L).
A full characterization of star height preserving homomorphisms was estab-lished later in [15], which reads as follows:
Theorem 7 (Hashiguchi/Honda). A homomorphism ρ : Γ∗→ Σ∗ preserves
star height if and only if (1) ρ is injective, (2) ρ is both prefix-free and suffix-free, that is, no word in ρ(Γ ) is prefix or suffix of another word in ρ(Γ ), and (3) ρ has the non-crossing property, that is, for all x1y1 and x2y2 that form two
distinct words in ρ(Γ ), at least one of the cross-wise concatenations x1x2 and
x2y1 do not belong to ρ(Γ ).
Observe that the given lower bound matches the aforementioned upper bound on the problem under consideration.
Theorem 8. There exists an infinite family of languages Ln over a binary
al-phabet Σ with alph(Ln) = O(n), such that alph(Σ∗\ Ln) = 22
Ω(n)
.
Proof. We will first prove the theorem for alphabet size 3, and then use a star-height preserving homomorphism to further reduce the alphabet size to binary. Let W2n be the set of walks in a complete 2n-vertex digraph as defined in
Theorem 5. Let E = { aij | 0 ≤ i, j ≤ 2n− 1 } denote the edge set of this graph,
and let Σ = {0, 1, $}.
Now define the homomorphism ρ : E∗ → Σ∗ by ρ(aij) = bin(i) · bin(j) ·
number i. Observe that ρ is star height preserving. To his end one has to verify the properties of Theorem 7. It is easily seen that the encoding is injective, and since ρ maps every symbol to a word of length 4n + 1, it is both prefix-free and suffix-free. Finally, for the noncrossing property, observe that the set ρ(E) coincides with the set of squares of length 4n over {0, 1} followed by a dollar sign, in symbols ρ(E) = {w2$ | w ∈ {0, 1}∗,|w| = 2n}. Assume x1y1 and x2y2
are two distinct square words of length 4n. If x1 and y1 have different lengths,
then the length of x1y2$ is not equal to 4n + 1. Otherwise, there is some position
j where x1y1 and x2y2 have different letters. Then in the word x1y2, the letter
at its j-th position differs from the letter at its (4n − j)-th position, and hence this word is not a square. Thus, ρ is a star height preserving homomorphism.
Our witness language for ternary alphabets is the complement of the set Ln= ρ(W2n). To establish the theorem for ternary alphabets, we give a regular
expression of size O(n) describing the complement of Ln; a lower bound of 22
Ω(n)
then immediately follows from Theorems 1 and 5 since the homomorphism ρ preserves star height. As for the witness language given in [8], our expression is a union of some local consistency tests: Every nonempty word in Lnfalls apart
into blocks of binary digits of each of length 4n, separated by occurrences of the symbol $, and takes the form
(bin(i0) bin(i1))2$ (bin(i1) bin(i2))2$ · · · $(bin(ir−1) bin(ir))2$.
Thus, word w is not in Ln if and only if we have at least one of the following
cases: (i) The word w has no prefix in {0, 1}4n$, or w contains an occurrence of $ not immediately followed by a word in {0, 1}4n$; (ii) the region around the
boundary of some pair of adjacent blocks in w is not of the form bin(i)$ bin(i); or (iii) some block does not contain the pattern (bin(i) bin(j))2, in the sense that inside the block some pair of bits at distance 2n does not match. To complete the proof for ternary alphabets, observe that at least one of the above three cases applies if and only if w matches the following regular expression of size O(n): rn= (0 + 1)≥1+ (ǫ + Σ∗$)¡(0 + 1)≤4n−1$ + (0 + 1)≥4n+1¢ Σ∗ + (Σ∗$(0 + 1)3n(0 + 1)∗0Σn+11Σ∗) + (Σ∗$(0 + 1)3n(0 + 1)∗1Σn+10Σ∗) + (Σ∗$(0 + 1)∗0Σ2n1Σ∗) + (Σ∗$(0 + 1)∗1Σ2n0Σ∗)
To further decrease the alphabet size to binary, we use the star height pre-serving homomorphism σ given in Theorem 6, which already proved useful in [10]. Then σ(Ln) has star height 2n and thus again has alphabetic width
at least 22Ω(n)
. For an upper bound on the alphabetic width of its complement, note first that every word w that is in σ(Σ∗) but not in σ(Ln) matches the
morphic image under σ of the expression rn given above; and σ(rn) still has
alphabetic width O(n). The words in the complement of σ(Ln) not covered by
the expression σ(rn) are precisely those not in σ({0, 1, $}∗), and the
The union of these two expressions gives a regular expression of size O(n) as
desired. ⊓⊔
3.3 Regular Expressions with Intersection and Interleaving
It is known that extending the syntax with an intersection operator can provide an exponential gain in succinctness over nondeterministic finite automata. For instance, in [6] it is shown that the set of palindromes of length n can be de-scribed by regular expressions with intersection of size O(n). On the other hand, it is well known that the number of states of a nondeterministic finite automaton accepting Pn has Ω(2n) states [21]. Of course, it appears more natural to
com-pare the gain in succinctness of such extended regular expressions to ordinary regular expressions rather than to finite automata. There a 22O(n)
doubly ex-ponential upper bound readily follows by combining standard constructions [7]. Yet a roughly doubly-exponential lower bound of 22Ω(√n), for alphabets of grow-ing size, was found only recently in [8], and a follow-up paper [7] shows that this can be reached already for binary alphabets. Here we finally establish a tight doubly-exponential lower bound, which even holds for binary alphabets. Theorem 9. There is an infinite family of languages Ln over a binary
al-phabet admitting regular expressions with intersection of size O(n), such that alph(Ln) = 22
Ω(n)
.
Proof. First, we show that the set of walks W2n ⊆ E∗ defined in Theorem 5
al-lows a compact representation using regular expressions with intersection. First we define M = { ai,j · aj,k | 0 ≤ i, j, k ≤ 2n− 1 } and the observe, that the set
Even of all nonempty walks of even length, i.e., total number of seen edges, in the graph Jn can be written as Even = M∗ ∩ (E · M∗ · E), while the the
set Odd of all nonempty walks of odd length is Odd = (E · M∗) ∩ (M∗· E). Thus, we have W2n = Even ∪ Odd ∪ {ǫ}. This way of describing W2n appears to
be a long shot from our goal; it uses a large alphabet and does not even reach a linear-exponential gain in succinctness over ordinary regular expressions—a similar statement appears, already over thirty years ago, in [4].
In order to get the desired result, we present a binary encoding τ that pre-serves star height and allows a representation of the encoded sets τ (M ) and τ (E) by regular expressions with intersection each of size O(n). Let τ : E∗→ {0, 1}∗ be the homomorphism defined by τ (ai,j) = bin(i) · bin(j) · bin(j)R· bin(i)R,
for 0 ≤ i, j ≤ 2n−1. To see that τ preserves star height, we have to check the properties given in Theorem 7. It can be readily seen that τ is injective, and it is both prefix-free and suffix-free, since all words in τ (E) are of the same length. The set τ (E) is just the set of binary palindromes of length 4n, and, by chance, a proof of the non-crossing property of this set is given already in [9], albeit in a different context. Thus, by Theorems 1 and 5, the set τ (W2n) has alphabetic
width at least 22Ω(n).
It remains to give expressions with intersection of size O(n) for the set τ (W2n).
Since τ (W2n) = τ (Even) ∪ τ (Odd) ∪ {ǫ}, the homomorphism commutes with
concatenation, union, and Kleene star, and, being injective, also with intersec-tion, it suffices to give regular expressions with intersection for τ (E) and τ (M )
of size O(n). To this end, we we make use of an observation from [6], namely that the sets of palindromes of length 2m admit regular expressions with in-tersection of size O(m). A straightforward extension of that idea gives a short regular expression with intersection for the set
Sm,n= { vwvR∈ {0, 1}∗ | |v| = m, |w| = n },
where m and n are fixed nonnegative integers: Namely, an expression rm,n
de-scribing this set is defined inductively by letting r0,n= (0 + 1)n
and
rm,n = ((0 + 1) · rm−1,n· (0 + 1)) ∩ (0(0 + 1)∗0 + 1(0 + 1)∗1) ,
for m > 0. Clearly, expression rm,n has size O(m + n) and describes the
lan-guage Sm,n. Next, observe that the set τ (E) = { wwR ∈ {0, 1}∗ | |w| = 2n }
is described by expression rn,0, which is of size O(n). Finally, note that the
set τ (M ) being equal to
{ bin(i) bin(j) bin(j)Rbin(i)Rbin(j) bin(k) bin(k)Rbin(j)R| 0 ≤ i, j, k ≤ 2n−1 }, can be written as τ (E)2∩{0, 1}2n·S
n,n·{0, 1}3n. The latter set can be described
by a regular expression with intersection of size O(n) again, and the proof is
completed. ⊓⊔
The interleaving of languages is another natural language operation known to preserve regularity. Regular expressions extended with interleaving were first studied in [18], with focus on the computational complexity of word prob-lems. They also showed that regular expressions extended with an interleaving operator can be exponentially more succinct than nondeterministic finite au-tomata [18]. Very recently, it was shown in [7] that regular expressions with interleaving can be roughly doubly-exponentially more succinct than regular expressions: Converting such expressions into ordinary regular expressions can cause a blow-up in required expression size of 22Ω(√n), for constant alphabet size. This bound is close to an easy upper bound of 22O(n)
that follows from standard constructions, see, e.g., [7] for details. If we take alphabets of growing size into account, the lower bound can be increased to match this trivial upper bound. The language witnessing that bound is in fact of very simple structure.
Theorem 10. There is an infinite family of languages Ln over an alphabet of
size O(n) having regular expressions with interleaving of size O(n), such that alph(Ln) = 22
Ω(n)
.
Proof. We consider the language Lndescribed by the shuffle regular expression
rn= (a1b1)∗ x (a2b2)∗ x · · · x (anbn)∗
To give a lower bound on the alphabetic width of Ln, we estimate first the
star height of Ln. The language Lncan be accepted by a 2n-state partial
bideter-ministic finite automaton A = (Q, Σ, δ, q0, F), whose underlying digraph forms
a symmetric n-dimensional hypercube: The set of states is Q = {0, 1}n, the state q0= 0n is the initial state, and is also the only final state, i.e., F = {0n}.
For 1 ≤ i ≤ n, the partial transition function δ is specified by δ(p, bi) = q and
δ(q, ai) = p, for all pairs of states (p, q) of the form (x0y, x1y) with x ∈ {0, 1}i−1
and y ∈ {0, 1}n−i. It can be readily verified that this partial deterministic finite
automaton is reduced and bideterministic. Therefore, the star height of Ln
coin-cides with the cycle rank of the n-dimensional symmetric Cartesian hypercube. For a symmetric graph G, the cycle rank of G coincides with its (undirected) elimination tree height, which is in turn bounded below by the (undirected) pathwidth of G. Many structural properties of the n-dimensional hypercube are known, and among these is the recently established fact [1] that its pathwidth equals Pn−1
i=0
¡ i
⌈i/2⌉¢ = Θ(2n−1/2 log n), where the latter estimate uses Stirling’s
approximation. Using Theorem 1, we obtain alph(Ln) = 2Ω(2n−1/2 log n)= 22
Ω(n)
,
as desired. ⊓⊔
For a similar result using binary alphabets, we will encode the above witness language in binary using a star height preserving homomorphism. Some extra care has to be taken, however. The ideal situation one might hope for is to find for each Γ = {a1, a2, . . . an} a suitable star height preserving homomorphism
ρ : Γ∗ → {0, 1}∗ such that ρ(x x y) = ρ(x) x ρ(y), for all x, y ∈ Γ∗. This
aim however appears to be a bit too ambitious. In all cases we have tried, the right-hand side of the above equation can contain words which are not even valid codewords. In [7] this difficulty is avoided altogether by simulating regular expressions with intersection by those with interleaving, using a trick from [18]. The drawback here is that the simulation takes place at the expense of introducing an extra symbol and polynomially increased size of the resulting expression with interleaving. To overcome this difficulty, Warmuth and Haussler devised a particular encoding [22], which they called shuffle resistant, that has the above property once we restrict our attention to codewords. Inspired by a property of this encoding proved later by Mayer and Stockmeyer [18, Prop. 3.1], we are led to define in general a shuffle resistant encoding as follows:
Definition 11. An injective homomorphism ρ : Γ∗ → Σ∗, for some alpha-bets Γ and Σ, is shuffle resistant if ρ(L(r)) = L(ρ(r)) ∩ ρ(Γ )∗, for each regular
expression r with interleaving over Γ .
The following is proved in [18, Prop. 3.1] for the encoding proposed by Warmuth and Haussler in [22]:
Theorem 12. Let Γ = {a1, a2, . . . , an} and Σ = {a, b}. The homomorphism
ρ: Γ∗ → Σ∗, which maps a
i to ai+1bi is shuffle resistant.
Incidentally, this encoding also preserves star height. The drawback is, how-ever, that alph(h(r)) = Θ(|Σ| alph(r)), for r a regular expression with interleav-ing. We now present a general family of more economic encodings, into alphabets of size at least 3, that enjoy similar properties.
Theorem 13. Let Γ and Σ be two alphabets, and $ be a symbol not in Σ. If ρ: Γ∗ → (Σ ∪ {$})∗ is an injective homomorphism with ρ(Γ ) ⊆ Σk$, for some
integer k, then ρ is shuffle resistant.
Proof. We need to show that for each such homomorphism ρ, the equality ρ(L(r)) = L(ρ(r)) ∩ ρ(Γ )∗ holds for all regular expressions r with interleaving
over Γ . The outline of the proof is roughly the same as the proof for Theorem 12 as sketched in [18]. The proof is by induction on the operator structure of r, using the stronger inductive hypothesis that
L(ρ(r)) ⊆ ρ(L(r)) ∪ E, with E = (ρ(Γ ))∗Σ≥k+1(Σ ∪ $)∗ (1) Roughly speaking, the “error language” E specifies that the first error occurring in a word in L(ρ(r)) but not in (ρ(Γ ))∗ must consist in a sequence of too many consecutive symbols from Σ.
The base cases are easily established, and also the induction step is easy for the regular operators concatenation, union, and Kleene star. The more difficult part is to show that if two expressions r1 and r2 satisfy Equation (1), then this
also holds for r = r1 x r2. To prove this implication, it suffices to show the
following claim:
Claim 14. For all words u, v in ρ(Γ )∗∪E and for each word z in u x v the
follow-ing holds: If both z ∈ (Σk$)∗ and u, v ∈ ρ(Γ )∗, then z ∈ ρ¡ρ−1(u) x ρ−1(v)¢.
Otherwise, z ∈ E.
Proof. We prove the claim by induction on the length of z. The base case with |z| = 0 is clear. For the induction step, assume |z| > 0 and consider the prefix y consisting of the first k + 1 letters of z. Such a prefix always exists if z is obtained from shuffling two nonempty words from ρ(Γ )∗∪ E. The cases where uor v is empty are trivial.
Observe first that it is impossible to obtain a prefix in Σ<k$ by shuffling two
prefixes u′ and v′ of the words u and v. Also, a prefix in Σ>k always completes to a word z ∈ E.
It remains to consider the case z has a prefix y in Σk$. To obtain such a
prefix, two prefixes u′ and v′ have to be shuffled, with (u′, v′) ∈ (Σj) × (Σk−j$) or (u′, v′) ∈ (Σj$) × (Σk−j). But since these are prefixes of words in ρ(Γ )∗∪ E, the index j can take on only the values j = 0 and j = k. Thus, if y ∈ Σk$,
then y is indeed in ρ(Γ ), and y is obtained by observing exclusively the first k+ 1 letters of u, or exclusively the first k + 1 letters of v. Hence at least one of the subcases y−1z∈ (y−1u) x v and y−1z∈ u x (y−1v) holds. We only consider
the first subcase, for the second one a symmetric argument applies.
It is not hard to see that we can apply the induction hypothesis to this subcase: Because y ∈ ρ(Γ ) and u ∈ ρ(Γ )∗ ∪ E, the word y−1u is again in the set ρ(Γ )∗∪ E. Having furthermore |y−1z| < |z|, the induction hypothesis
readily implies that claimed statement also holds for the word z = y(y−1z). This completes the proof of the claim. ⊓⊔ Having established the claim, completing the proof of the statement L(ρ(r)) ⊆ ρ(L(r)) ∪ E is a rather easy exercise. ⊓⊔
The existence of economic shuffle resistant binary encodings that further-more preserve star height is shown next.
Theorem 15. Let Γ be an alphabet. There exists a homomorphism ρ : Γ∗ → {0, 1}∗ such that (1) |ρ(a)| = O(log |Γ |), for every symbol a ∈ Γ , and (2) the
homomorphism ρ is shuffle resistant and preserves star height.
Proof. Without loss of generality assume Γ = {a1, a2, . . . a2k} for some k ≥ 0. In
a first step, we encode into an alphabet of size three. Let σ : Γ∗ → ({0, 1}∪{$})∗ be the homomorphism given by σ(ai) = bin(i) · bin(i)$, with bin(i) being the
usual k-bit binary encoding of the number i. Obviously, σ maps all alphabet symbols to strings of length O(log |Γ |). That the encoding is shuffle resistant follows from Theorem 13, and that it preserves star height is shown along the same lines as in the proof of Theorem 8, where we studied a rather similar encoding. In a second step, we use the homomorphism τ from Theorem 12 to further decrease the alphabet size from ternary to binary. This encoding is both shuffle resistant and preserves star height. The composed encoding ρ = τ ◦ σ does the job: We have τ (σ(a)) = O(log |Γ |), and it is readily proved that τ ◦ σ is both shuffle resistant and preserves star height by expanding the definitions
of these two notions. ⊓⊔
For regular expressions with interleaving we show that the conversion to ordinary regular expressions induces a 22Ω(n/ log n) lower bound for binary input alphabet.
Theorem 16. There is an infinite family of languages Ln over a binary
al-phabet admitting regular expressions with interleaving of size O(n), such that alph(Ln) = 22
Ω(n/ log n)
.
Proof. Our witness language will be described by the expression ρ(rn) = (ρ(a1)ρ(b1))∗ x (ρ(a2)ρ(b2))∗x · · · x (ρ(an)ρ(bn))∗,
obtained by applying the homomorphism ρ from Theorem 15 to the expression rn used in the proof of Theorem 10. This expression has size O(n log n), and to
prove the theorem, it will suffice to establish that L(ρ(rn)) has alphabetic width
at least 22Ω(n)
.
Recall from the proof of Theorem 10 that the star height of L(rn) is bounded
below by 2Ω(n). Since ρ preserves star height, the same bound applies to the language ρ(L(rn)). By Theorem 1, we thus have
alph(ρ(L(rn))) = 22
Ω(n)
. (2)
Unfortunately, this bound applies to the language ρ(L(rn)) rather than to
L(ρ(rn)). At least, as we know from Theorem 15 that ρ is a shuffle resistant
encoding, these two sets are related by
L(ρ(rn)) ∩ ρ(Γ )∗= ρ(L(rn)), (3)
To derive a similar lower bound on the language L(ρ(rn)), we use the
2O(n(1+log m)) upper bound from [12] on the alphabetic width of the
intersec-tion for regular languages of alphabet width m and n, respectively, for m ≥ n. To this end, let α = α(n) denote the alphabetic width of L(ρ(rn)). We show first
that α(n) ≥ alph(ρ(Γ )∗). Assume the contrary. By Theorem 15, the set ρ(Γ )∗
admits a regular expression of size O(n log n). Assuming α(n) ≤ alph(ρ(Γ )∗), the upper bound on the alphabetic width of intersection implies that ρ(L(rn)) =
L(ρ(rn)) ∩ ρ(Γ∗) admits a regular expression of size 2O(n log
2n)
. But this clearly contradicts Inequality (2). Thus, α ≥ alph(ρ(Γ )∗). Applying the upper bound
for intersection to the left-hand side of Equation (3), we obtain
alph(ρ(L(rn))) = alph(L(ρ(rn)) ∩ ρ(Γ∗)) = 2O(n log n log α). (4)
Inequalities (2) and (4) now together imply that there exist positive constants c1
and c2 such that, for n large enough, holds 22c1n ≤ 2c2n log n log α. Taking double
logarithms on both sides and rearranging terms, we obtain c1n− O(log n) ≤
log log α. Since the the left-hand side is in Ω(n), we thus have alph(L(ρ(rn))) =
α= 22Ω(n)
, and the proof is completed. ⊓⊔
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